Martians

A sphere-shaped spaceship with a diameter of 6 m landed in the meadow. In order not to attract attention, the Martians covered it with a roof in the shape of a regular cone. How high will this roof be so that the consumption of roofing is minimal?

Correct answer:

S =  115.6965 m²

Step-by-step explanation:

$$\begin{gathered} D=6\text{ m } \\ R=D\mathrm{/}2=6\mathrm{/}2=3\text{ m } \\ \sin \alpha =r\mathrm{/}s \\ R:(h-R)=r:s=\sin \alpha \\ r=s\cdot {\displaystyle \frac{R}{h-R}} \\ S=\pi \cdot r\cdot s=\pi \cdot R\mathrm{/}(h-R)\cdot s^2 \\ S=\pi \cdot R\mathrm{/}(h-R)\cdot ((R\mathrm{/}(h-R){)}^2+h^2) \\ S=(3(9\mathrm{/}(-3+h{)}^2+h^2)\pi )\mathrm{/}(-3+h) \\ h=6.37252\doteq 6.3725\text{ m } \\ s=\sqrt{(R\mathrm{/}(h-R){)}^2+h^2}=\sqrt{(3\mathrm{/}(6.3725-3){)}^2+6.3725^2}\doteq 6.4343\text{ m } \\ r=s\cdot {\displaystyle \frac{R}{h-R}}=6.4343\cdot {\displaystyle \frac{3}{6.3725-3}}\doteq 5.7236\text{ m } \\ S=\pi \cdot r\cdot s=3.1416\cdot 5.7236\cdot 6.4343=115.6965{\text{m}}^2 \end{gathered}$$

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